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Heuristics is a procedure based on chemical graph theory that provides direction to solution of the problem. It primarily focus on the passage from molecular indices to molecular connectivity terms. This procedure includes construction of molecular connectivity terms, their linear combinations, and the changes required to model properties of different classes of compounds. It is easier than quantum mechanical methods and equally applicable to each class of compounds.

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... Moreover, quantum transport in multiply connected systems [11], like electron transport in organic molecules [12] as proteins and polymers, may be described by one-dimensional pathways (trajectories through the edges), changing from one path to another due to scattering at the vertices centers. More recently, quantum graphs have also been used to characterize molecular connectivity [13,14]. ...

... In the Appendix B we outline the main steps necessary to prove that the exact Green's function for arbitrary quantum graphs has the very same form of Eq. (13). Moreover, as we are going to discuss in length in Sec. 4, different techniques can be used to identify and sum up all the scattering paths. ...

... The formula in Eq. (13) gives the correct Green's function for arbitrary connected undirected simple quantum graphs. However, it has no universal practical utility unless we are able to identify all the possible scattering paths and to sum up the resulting infinite series -regardless the specific system. ...

Here we review the many aspects and distinct phenomena associated to quantum dynamics on general graph structures. For so, we discuss such class of systems under the energy domain Green’s function (G) framework. This approach is particularly interesting because G can be written as a sum over classical-like paths, where local quantum effects are taken into account through the scattering matrix elements (basically, transmission and reflection amplitudes) defined on each one of the graph vertices. Hence, the exact G has the functional form of a generalized semiclassical formula, which through different calculation techniques (addressed in detail here) always can be cast into a closed analytic expression. It allows to solve exactly arbitrary large (although finite) graphs in a recursive and fast way. Using the Green’s function method, we survey many properties of open and closed quantum graphs as scattering solutions for the former and eigenspectrum and eigenstates for the latter, also considering quasi-bound states. Concrete examples, like cube, binary trees and Sierpiński-like topologies are presented. Along the work, possible distinct applications using the Green’s function methods for quantum graphs are outlined.

... In analogy to the molecular vector x used to represent organic molecules (Marrero- single excitation energies in eV, and first (f 1 ) and second (f 2 ) oscillator strength values (of the first singlet excitation energies) of the nucleotide DNA-RNA bases, and so on (Pogliani, 2000). For instance, the f 1 (B) property of the DNA-RNA bases B takes the values f 1 = 0.28 for adenine, f 1 (G) = 0.20 for guanine, f 1 (U) = 0.18 for uracile, and so on (Pogliani, 2000). ...

... In analogy to the molecular vector x used to represent organic molecules (Marrero- single excitation energies in eV, and first (f 1 ) and second (f 2 ) oscillator strength values (of the first singlet excitation energies) of the nucleotide DNA-RNA bases, and so on (Pogliani, 2000). For instance, the f 1 (B) property of the DNA-RNA bases B takes the values f 1 = 0.28 for adenine, f 1 (G) = 0.20 for guanine, f 1 (U) = 0.18 for uracile, and so on (Pogliani, 2000). Table 1 depicts nucleotides (bases) descriptors properties for DNA-RNA bases. ...

... Finally, the satisfactory comparative results showed that nucleic acid bilinear indices used here will be a novel chem-and bioinformatics tool for further research. 0.13 0.72 7.5 4.61 6.26 Experimental molar absorption coefficient Є 260 at 260 nm and PH = 7.0, first (ΔE 1 ) and second (ΔE 2 ) single excitation energies in eV, and first (f 1 ) and second (f 2 ) oscillator strength values (of the first singlet excitation energies) of the nucleotide DNA-RNA bases (Pogliani, 2000). ...

... The multilinear least-squares (MLS) used to derive the quantitative structure-property relationships (QSPR) showed that three out of eleven properties, the refractive index (RI), the flash points (FP), and the ultraviolet cutoff values (UV), were modeled with the MMCI while the remaining properties were modeled with the well-known molecular connectivity indices (MCI). The MMCI indices are also centered on the basic concepts of the delta, valence delta, I-and S-indices that go back to the origins of the molecular connectivity theory [2][3][4][5][6][7]. Results from two other recent studies that used semiempirical sets of descriptors [8,9] showed that the artificial neural network (ANN) model with a variable number of hidden neurons chosen by the software improves the quality of a QSPR obtained with the aid of the multilinear least-squares (MLS) methodology, also known as multilinear regression (MLR). ...

... Summing up, we have thirty-one MCI and thirty-six MMCI. Every index was obtained with a visual basic home-made program that runs on a normal PC that uses both adjacency and distance matrices [6]. ...

... Concerning the statistical results for the training compounds, ANN 1HN (Table 9, 1st line) improves over MLS for T b , and El properties, while it lays behind for −χ·10 6 , otherwise results are rather similar. With the whole set of compounds (Table 9, second line); i.e., with training (and test with ANN)-plus evaluated compound ANN 1HN calculations improve again over MLS for T b , and El, while they stay behind with ε, γ, UV, and −χ·10 6 . ...

The mean molecular connectivity indices (MMCI) proposed in previous studies are used in conjunction with well-known molecular connectivity indices (MCI) to model eleven properties of organic solvents. The MMCI and MCI descriptors selected by the stepwise multilinear least-squares (MLS) procedure were used to perform artificial neural network (ANN) computations, with the aim of detecting the advantages and limits of the ANN approach. The MLS procedure can replicate the obtained results for as long as is needed, a characteristic not shared by the ANN methodology, which, on the one hand increases the quality of a description, and on the other hand also results in overfitting. The present study also reveals how ANN methods prefer MCI relatively to MMCI descriptors. Four types of ANN computations show that: (i) MMCI descriptors are preferred with properties with a small number of points, (ii) MLS is preferred over ANN when the number of ANN weights is similar to the number of regression coefficients and, (iii) in some cases, the MLS modeling quality is similar to the modeling quality of ANN computations. Both the common training set and an external randomly chosen validation set were used throughout the paper.

... By considering atomic orbitals and electrons as vertices and overlap between them as edges, a wealth of knowledge has been extracted on electron delocalized molecules [5][6][7][8]. Chemical graphs are the easiest way to keep us floating in the rising sea of experimental data, by compressing them into algorithms [9]. Computers are then effectively used to obtain solutions to such large chemical graphs, by implementing the particular algorithm. ...

... As seen from the table, the largest node is λ 9 for man5c, man5h and min5h and node λ 10 is for man6c, man6h and min6h. The chemical structures and nomenclature of the target radicals in Figure 1 reveals that C (9) is the bridge carbon atom in the COO(H) group of the corresponding Set I radicals, whereas C (10) is the bridge carbon atom in the COO(H) group for the Set II radicals. Interestingly, in both Set I and Set II, min5c and min6c radicals have C (2) and C (4) with degenerate λ 2 and λ 4 , indicating that the unpaired electron of the radicals (see Figure 1) accumulates sufficient electron density to form a pseudo-conjugate local structure at the top section of the T-shaped min5c and min6c. ...

... Finally, the depiction of core structure of alkyl radicals in the form of degree matrix (DM) [42] in Figure 5 captures the important nodes of the T-junctions of the target radicals, C (1) and C (9) , respectively. This position C (1) with DM value of 3 indicates a three-way connection at this atom and locates the unpaired electron of the radicals and therefore, the most important chemical spot of radicals. ...

Alkyl radicals play important roles such as intermediary metabolism, cell damage/injury and death leading to potential mutations. The present investigation using chemical graph theory studied two sets of carboxyl radicals, that is, deprotonated (DPro, radical) and protonated (Pro, radical anion) forms of 5‐ethyl heptanoic acid and 5‐ethenyl hept‐6‐enoic acid (Set I) radicals and 6‐ethyl octanoic acid and 6‐ethenyl oct‐7‐enoic acid (Set II) radicals. The study reveals that the largest eigenvalue (LEV) spectra of the adjacency matrix have a unique value, where the spectra increase from DPro to Pro and from single to double bonded alkyl radical structures, thus forming a scoring function for molecular topological indices. This topological index is presented as a measure for molecular connectivity/branching, where the index is used to predict the refractivity of a series of carboxyl radicals. The statistical correlation coefficient obtained for quantitative structure–property relationship between chemical structure (alkyl radical) and its physical property (refractivity) through heat maps are excellent and ranges within 0.97–0.99. It is further discovered that the vector component of the LEV gives an insight to its structural details, where it captures the node with the highest degree along with the important weighted node, that holds the complete structure (i.e., the radical site), in case of the Pro radical structures. Node centrality, which captures the structural makeup, divides DPro radical T‐shaped structures into two subunits for the signal transduction of important biological process like oral toxicity. Size of the largest clusters is also studied, illustrating the parameter to be less sensitive for differentiating the CC double bonds in the Pro radicals. To our knowledge, this is the first study where pattern recognition has been exemplified through the lower‐diagonal‐upper decomposition matrix of the chemical graph that forms a fingerprint signature to differentiate the alkyl radicals. The present study innovatively digitalizes the chemical structures of alkyl radicals that enables the discovery of structure–property relationship reflected by their molecular branching through machine learning.

... This Index was first appeared in [5]. Zhong found the minimum and maximum values of the harmonic index for simple connected graph and trees, and characterized the corresponding extremal graph. ...

... . proof: Let G be a bipartite graph with partition | X |= m and | Y |= n. number of edges in G is mk1+nk2 2 In the Total graph of G, |V (T (G))| = m + n + mk1+nk2 (4,5), (4,6), (5,5), (5,6), (5,7), (6,6), (6,7), (6,8), (7,7), (7,8), (8,8) respectively. ...

... . proof: Let G be a bipartite graph with partition | X |= m and | Y |= n. number of edges in G is mk1+nk2 2 In the Total graph of G, |V (T (G))| = m + n + mk1+nk2 (4,5), (4,6), (5,5), (5,6), (5,7), (6,6), (6,7), (6,8), (7,7), (7,8), (8,8) respectively. ...

Let G = G(V,E) be a graph with | V | vertices and | E | edges and total graph, T(G) is obtained from G. In This paper we have study the Harmonic index of total graph for standards graphs, bipartite graph of particular type, regular graph, Grid graph Gm,n and Complete Binary tree.

... In 1947, the chemist Wiener opened a new horizon for the chemists and mathematicians by using a topological index to study the relation between molecular structure and physical and chemical properties of certain hydrocarbon compounds [4]. Since 1947, more than hundred topological indices have been defined, from which, Randić connectivity index is one of the most successful molecular descriptors in structureproperty and structure-activity relationships studies [5,6,7]. The Randić connectivity index [8] or product-connectivity index [9,10] ...

... Using inequality (6) in equation (5), we have ...

Graph theory is nowadays a powerful tool in predicting a large number
of physico-chemical properties of chemical compounds. There are
various methods to quantify the molecular structures, of which the
topological index is the most popular since it can be obtained directly
from molecular structures and rapidly computed for large number
of molecules. Among these indices, the Randić connectivity index
is the most successful and commonly used degree-based molecular
descriptor in structure-property and structure-activity relationships
studies. In this paper, the lower and upper bounds for general Randić,
general sum-connectivity and harmonic indices for tensor product of
certain graph operations on graphs are determined.

... where d(u) denotes the degree of a vertex u of the graph G and uv the edge connecting the vertices u and v. Randić noticed that this index was well correlated with a variety of physico-chemical properties of alkanes: boiling point, enthalpy of formation, surface area, solubility in water, etc. Eventually, this index became one of the most successful molecular descriptors in structure property and structure activity relationship studies [40,55,56,72], and scores of its pharmacological and chemical applications have been reported. Mathematical properties of this descriptor have also been studied extensively, as summarized Gutman [43,57]. ...

... This index became one of the most successful molecular descriptors in structure property and structure activity relationship studies [40,55,56,72,76], and scores of its pharmacological and chemical applications have been reported. Mathematical properties of this descriptor have also been studied extensively, see [43,57]. ...

... In 1975, Randić introduced a moleculor quantity of branching index [1], which has been known as the famous Randić connectivity index and that is a most useful structural descriptor in QSPR and QSAR, see [2,3,4,5]. Mathematicians have considerable interests in the structural and applied aspects of Randić connectivity index, see [6,7,8,9]. Based on the successful considerations, Zagreb indices [10] are introduced as an expected formula for the total π-electron energy of conjugated molecules as follows. ...

For a graph $G$, the first multiplicative Zagreb index $\prod_1(G) $ is the product of squares of vertex degrees, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of products of degrees of pairs of adjacent vertices. In this paper, we explore graphs with extremal $\Pi_{1}(G)$ and $\Pi_{2}(G)$ in terms of (edge) connectivity and pendant vertices. The corresponding extremal graphs are characterized with given connectivity at most $k$ and $p$ pendant vertices. In addition, the maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ are provided. Our results extend and enrich some known conclusions.

... We use Wiener index for finding correlation between molecular structure of certain hydrocarbon compounds and their physical and chemical properties . Since 1947, more than hundred topological indices have been defined, from which, Randić connectivity index is listed among the most useful molecular descriptors in structure-activity and structure-property relationships studies [23,27,30]. The Randić index [24] of the graph G, also known as product-connectivity index [21,32] is defined as: ...

Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In this paper, bounds for the Randić, general Randić, sum-connectivity, the general sum-connectivity and harmonic indices for tensor product of graphs are determined by using the combinatorial inequalities and combinatorial computing.

... Later, branching index was named as Randić index. Randić index is one of the most used topological indices in QSAR and QSPR studies, for example, see [2][3][4][5][6][7]. Mathematicians also exhibited significant interest in the properties of the Randić connectivity index, for example, see [8][9][10][11][12][13][14][15][16]. ...

A connected graph G is said to be cactus if no two cycles of G have any common edge. The present note is devoted to developing some extremal results for the zeroth-order general Randic index of cactus graphs and finding some sharp bounds on this index.

... Thus, it is possible that hydrogen bonds suppressed in the chemical graph structure for molecular connectivity calculations could be considered a key parameter in this research. As a result, they should be included in any discussion of molecular connectivity indicators for new antimalarial drug candidates [70]. ...

With the aim of researching new antimalarial drugs, a series of quinoline, isoquinoline, and quinazoline derivatives were studied against the Plasmodium falciparum CQ-sensitive and MQ-resistant strain 3D7 protozoan parasite. DFT with B3LYP functional and 6-311G basis set was used to calculate quantum chemical descriptors for QSAR models. The molecular mechanics (MM2) method was used to calculate constitutional, physicochemical, and topological descriptors. By randomly dividing the dataset into training and test sets, we were able to construct reliable models using linear regression (MLR), nonlinear regression (MNLR), and artificial neural networks (ANN). The determination coefficient values indicate the predictive quality of the established models. The robustness and predictive power of the generated models were also confirmed via internal validation, external validation, the Y-randomization test, and the applicability domain. Furthermore, molecular docking studies were conducted to identify the key interactions between the studied molecules and the PfPMT receptor’s active site. The findings of this contribution study indicate that the antimalarial activity of these compounds against Plasmodium falciparum appears to be largely determined by four descriptors, i.e., total connectivity (Tcon), percentage of carbon (C (%)), density (D), and bond length between the two nitrogen atoms (Bond N–N). On the basis of the reliable QSAR model and molecular docking results, several new antimalarial compounds have been designed. The selection of drug candidates was performed according to drug-likeness and ADMET parameters.

... QSAR-2D: correlacionan la actividad con patrones estructurales de las moléculas, teniendo en consideración la topología de las mismas. Cuando se habla de topología molecular es en referencia a la consideración de la molécula como un conjunto de vértices (los átomos) unidos por aristas (los enlaces) denominado grafo (Pogliani, 2000). La información que puede extraerse con esta metodología es por demás variada y va desde índices de conectividad molecular sencillos hasta complejos cálculos vectoriales realizados sobre matrices construidas a partir de los grafos (Todeschini & Consonni, 2000). ...

... One of the most well-known topological indices is called Randić index, a moleculor quantity of branching index [1]. It is known as the classical Randić connectivity index, which is the most useful structural descriptor in QSPR and QSAR, see [2,3,4,5]. Many mathematicians focus considerable interests in the structural and applied issues of Randić connectivity index, see [6,7,8,9]. ...

The first Zagreb index of a graph $G$ is the sum of the square of every vertex degree, while the second Zagreb index is the sum of the product of vertex degrees of each edge over all edges. In our work, we solve an open question about Zagreb indices of graphs with given number of cut vertices. The sharp lower bounds are obtained for these indices of graphs in $\mathbb{V}_{n,k}$, where $\mathbb{V}_{n, k}$ denotes the set of all $n$-vertex graphs with $k$ cut vertices and at least one cycle. As consequences, those graphs with the smallest Zagreb indices are characterized.

... The Randić index is one of the most successful molecular descriptors in the studies of quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR). In view of its successful applications in QSPR and QSAR, some researchers gave physicochemical interpretation of this molecular descriptor [1][2][3][4][5]. The Randić index has many chemical applications and became a popular topic of research in mathematics and mathematical chemistry [6,7]. ...

The relationship between Randić index, sum-connectivity index, harmonic index and pi-electron energy of some benzenoid hydrocarbons is obtained.

... Such new synthesis is possible by the correlation analysis, in that correlation is investigated by correlating the activity with topological indices5. Needless to state that topological index is a numerical representation of molecular structure [5][6]. Hence, quantitative structureactivity relationship (QSAR) is only possible using topological indices as molecular descriptors. ...

... Many variations of the Randić connectivity index have been discussed in different studies. [2][3][4][5][6][7][8][9] The second connectivity index is expressed as ...

The Randi (product) connectivity index and its derivative called the sum-connectivity index are well-known topological indices and both of these descriptors correlate well among themselves and with the π-electronic energies of benzenoid hydrocarbons. The general n connectivity of a molecular graph G is defined as n (G)=-v i 1 v i 2 v i 3. v i n+1 (1/d i 1 d i 2. d i n+1) and the n sum connectivity of a molecular graph G is defined as n X(G)=- v i 1 v i 2 v i 3.v i n+1 (1/d i 1 +d i 2 +.+d i n+1), where the paths of length n in G are denoted by v i 1, v i 2,.,v i n+1 and the degree of each vertex vi is denoted by di. In this paper, we discuss third connectivity and third sum-connectivity indices of diamond-like networks and compute analytical closed results of these indices for diamond-like networks.

... In 1975, Randić introduced a moleculor quantity of branching index [1], which has been known as the famous Randić connectivity index that is most useful structural descriptor in QSPR and QSAR, see [2,3,4,5]. Mathematicians have considerable interests in the structural and applied issues of Randić connectivity index, see [6,7,8,9]. Based on the successful considerations, Zagreb indices [10] are introduced as an expected formula for the total π-electron energy of conjugated molecules as follows. ...

For a (molecular) graph, the first multiplicative Zagreb index $\prod_1(G) $ is the product of the square of every vertex degree, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore graphs in terms of (edge) connectivity. The maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ of graphs with connectivity at most $k$ are provided. In addition, the corresponding extremal graphs are characterized, and our results extend and enrich some known conclusions.

... The Randić index R(G), proposed by Randić [12] in 1975, one of the most used molecular descriptors in structure-property and structure-activity relationship studies [7,8,10,11,13,14,16], was defined as ...

Du, Zhou and Trinajstic [J. Math. Chem., 48, pp. 697-703, 2010] proved that in the set of n-vertex connected unicyclic graphs G with n >= 3, for 1 <= alpha < 0, the graph consisting of a triangle and n 3 pendant vertices adjacent to the same vertex of this triangle is the unique graph with the minimum general sum-connectivity index. In this paper, using a graph transformation and several inequalities, we show that in the class of n-vertex connected bicyclic graphs G with n >= 4, the graph consisting of two triangles having a common edge and n 4 pendant vertices adjacent to the same vertex of degree three of this graph is the unique graph minimizing the general sum-connectivity index for -1 <= alpha < 0.

... The Randić index [20] R(G) is one of the most successful molecular descriptors in the studies of structure-property and structureactivity relationships [6,10,11,19] and is defined as ...

The harmonic index of a graph G is defined as the sum of weights $\frac{2}{d_G(u)+d_G(v)}$ of all edges uv of G and the Randic index of a graph G is defined as the sum of weights $\frac{1}{\sqrt{d_G(u)d_G(v)}}$ of all edges uv of G, where $d_G(u)$ is the degree of a vertex u in G. In this paper, the expressions for the harmonic index and Randic index of the generalized transformation graphs $G^{xy}$ and for their complement graphs are obtained in terms of the parameters of underline graphs.

... In chemistry, for a molecule, we can get a molecular graph by representing the atoms by vertices and the bonds by edges [27,31]. Graph invariants of molecular graph, that predict the corresponding molecule properties, are called topological indices. ...

Let G be a vertex-weighted graph with vertex-weighted function ω:V(G)→R+ satisfying ω(vi)=xi for each vi∈V(G)={v1,v2,…,vn}. The weighted Wiener index of vertex-weighted graph G is defined as W(G;x1,x2,…,xn)=∑1≤i<j≤nxixjdG(vi,vj), where dG(vi,vj) denotes the distance between vi and vj in G. In this paper, we give a combinatorial explanation of W(G;x1,x2,…,xn) when G is a tree: W(G;x1,x2,…,xn)=m(S(G),n−2)∏i=1nxi, where m(S(G),n−2) is the sum of weights of matchings of some weighted subdivision S(G) of G with n−2 edges. We also give a similar combinatorial explanation of the weighted Kirchhoff index K(G;x1,x2,…,xn)=∑1≤i<j≤nxixjrG(vi,vj) of a unicyclic graph G, where rG(vi,vj) denotes the resistance distance between vi and vj in G. These results generalize some known formulae on the Wiener and Kirchhoff indices.

... The famous Randić index [18], has been used effectively as a molecular descriptor in quantitative structurepharmacokinetics relationship (QSPR) and quantitative structure-activity relationship (QSAR) (see Pogliani [19], García-Domenech et al. [20], Stankevich et al. [21], Galvez [22], Estrada [23], Klein et al. [24]). If G is a graph then its Randić index is denoted as: ...

A large number of medical experiments have confirmed that the features of drugs have a close correlation with their molecular structure. Drug properties can be obtained by studying the molecular structure of corresponding drugs. The calculation of the topological index of a drug structure enables scientists to have a better understanding of the physical chemistry and biological characteristics of drugs. In this paper, we focus on Hyaluronic Acid-Paclitaxel conjugates which are widely used in the manufacture of anticancer drugs. Several topological indices are determined by virtue of the edge-partition method, and our results remedy the lack of medicine experiments, thus providing a theoretical basis for pharmaceutical engineering.

... All graphs taken into consideration here are simple, finite and undirected graphs. The Randić index due to [11], denoted by , G R also known as product connectivity index is defined as According to [4], [6], [7], [9] and [13], Randić index is the most successful molecular descriptor and finds applications in multitude of scenarios. ...

Predictive potential of Randić index, Sum-connectivity index and Harmonic index is studied. These topological indices are utilized as parameters while predicting the molecular weights (in , mol g densities (in , C boiling points (in C and melting points (in C of Benzenoid hydrocarbons. It is shown that these indices are good predictors of the molecular weight (in , mol g and boiling point (in . C However thy are not sufficient to predict melting point (in C and density (in , 3 cm g when used alone. Usefulness of additional information about the molecular structure of Benzenoid hydrocarbons in improving the performance of the molecular descriptors as predictors is also discussed.

... An unexpected mathematical quality of the Randić index was discovered recently, which tells us about the relation of this topological invariant with the normalized Laplacian matrix [38,40,41]. e GRI known as general RI [42] is defined as ...

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In this paper, M-polynomial OKn and OPn networks are computed. The M-polynomial is rich in information about degree-based topological indices. By applying the basic rules of calculus on M-polynomials, the first and second Zagreb indices, modified second Zagreb index, general Randić index, inverse Randić index, symmetric division index, harmonic index, inverse sum index, and augmented Zagreb index are recovered.

... In this paper, we computed first and second reverse Zagreb indices, first and second reverse hyper-Zagreb indices, reverse GA index, reverse atomic bond connectivity index, first and second reverse Zagreb polynomials, and first and second reverse hyper-Zagreb polynomials for the crystallographic structure of molecules [24,25]. Our results are important to guess the properties [26][27][28] and study the topology of the crystallographic structure of molecules and can be used in drug delivery [29][30][31]. ...

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. Topological indices help us collect information about algebraic graphs and give us mathematical approach to understand the properties of algebraic structures. With the help of topological indices, we can guess the properties of chemical compounds without performing experiments in wet lab. There are more than 148 topological indices in the literature, but none of them completely give all properties of under study compounds. Together, they do it to some extent; hence, there is always room to introduce new indices. In this paper, we present first and second reserve Zagreb indices and first reverse hyper-Zagreb indices, reverse GA index, and reverse atomic bond connectivity index for the crystallographic structure of molecules. We also present first and second reverse Zagreb polynomials and first and second reverse hyper-Zagreb polynomials for the crystallographic structure of molecules.

... Initially, the Randić index was studied only by chemists, but recently it has also attracted the attention of mathematicians, for instance, we cite [2,3,6]. The Randić index found chemical applications and became a popular topic of research in mathematics and mathematical chemistry, for more information you can see [23,26,27]. Randić proposed this index in order to quantitatively characterize the degree of molecular branching. ...

Let G be a graph with vertex set V (G) and edge set E(G), and let du
Pdenote the degree of vertex u 2 V (G). The Randi�c index of G is de�ned as R(G) =
uv2E(G) 1=
p
dudv: In this paper, we investigate the relationships between Randi�c
index and several topological indices.

... It is one of the most popular molecular invariants in the study of structure-property and activity relationships [3,4]. e general Randić index is described as follows: ...

The study of networks and graphs through structural properties is a massive area of research with developing significance. One of the methods used in studying structural properties is obtaining quantitative measures that encode structural data of the whole network by the real number. A large collection of numerical descriptors and associated graphs have been used to examine the whole structure of networks. In these analyses, degree-related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree-related indices is one of the successful topics of research. The general sum-connectivity GSC index of graph Q is described as χρQ=∑qq′∈EQdq+dq′ρ, where dq presents the degree of the vertex q in Q and ρ is a real number. The total graph TQ is a graph whose vertex set is VQ∪EQ, and two vertices are linked in TQ if and only if they are either adjacent or incident in Q. In this article, we study the general sum-connectivity index χρQ of total graphs for different values of ρ by using Jensen’s inequality.

... An inspection of the interrelation between the eight terms confirms their small interrelation as <r IM (pI:{X})> = 0.560, r w ( D X,X t ) = 0.004 and r s ( D X, 1 X) = 0.975, where r w and r s stand for the weakest and strongest interrelations, respectively. A critical analysis of the 0 X v term lets us notice that this term is trivial, as it is nothing other than (1 + Δn/n T ) [6]. Now as the best description is given by a relation consisting of only this term, this means that molecular connectivity indices are not needed to simulate this property. ...

Quantitative structure activity relationship (QSAR) studies were performed on a series of azole-based inhibitors of HIV-1 reverse transcriptase (RT). The antiviral activity of azole-based non-nucleoside inhibitors of HIV-1 RT were found to be influenced with electronic parameter - polarizability (Pz), steric parameters - molecular weight (Mw), molar refractivity (Mr), index of refraction (lor), surface tension (St), and topological parameters - the Balaban centric index (Bac), the Balaban distance connectivity index (J) and second and fourth-order connectivity indices ( 2X, 4x) in combination with some indicator parameters. The parameters - polarisability (Pz), molecular weight, molar refractivity, the Balaban centric index (Bac) and second and fourth-order connectivity indices ( 2X, 4X) were found to play positive role whereas the parameters index of refraction (lor), surface tension (St) and the Balaban distance connectivity index (J) were found to play negative role in determining the activity. The presence of CH 2(2'-Cl, 4'-ClPh) group at Reposition is conducive whereas phenyl and 3'-thiophenyl groups at Reposition and CH 2 (4'-C1Ph) group at Reposition are detrimental for the activity. The results are critically discussed on the basis of regression data and crossvalidation technique. The Pogliani factor Q and the results of leave-one-out (LOO) method confirm the reliability and predictability of the proposed models.

The homology (homolo) relation between molecules was introduced. This relation is a generalization of an old idea of series of homologous compounds. The homolo relation operates on a molecule by removing all the structural fragments that are identical with a certain selected fragment. As a result a multiset of fragments is produced. It was shown that the homolo relation is an equivalence relation in a set of molecules. Thus, by choosing various reference fragments, the molecules can be classified in many different ways. Using the language of homolo operation it is possible to redefine such ideas as constitutional and stereo isomers as well as a generator of a molecule and, for instance, factorization of a molecule onto fragments.

Nowadays, the increasing interest in zeolite synthesis from low-cost materials has promoted the development of various studies on their conversion into zeolitic materials, giving rise to an extensive literature. The enhancing demands for a healthy environment, particularly with regards to high quality drinking water and removal of pollutants from industrial, agricultural and municipal wastewater, are a major reason to search for new materials. Nearly every application of zeolites has driven by environmental concerns, or plays a significant role in reducing toxic waste and energy consumption. Because of their unique porous properties, zeolites are used in a variety of applications with a global market of several million tonnes per year. Major uses are in petrochemical cracking, ion-exchange (water softening and purification), and in the separation and removal of gases and solvents. Other applications are in construction, agriculture, animal husbandry, medicine, etc. A brief overview of the zeolite fields of industrial and environmental applications covering the fundamentals and recent developments is proposed.

Graph Theory (GT) and its applications in chemistry, the so-called Chemical Graph Theory (CGT), appear to be two of the most misunderstood areas of theoretical chemistry. We outline briefly possible causes for misunderstanding and suggest remedies, including a test on the knowledge of GT and CGT.

The original formulation of the Zagreb indices is presented and their relationship to topological indices such as self-returning walks, Platt, Gordon-Scantlebury and connectivity indices is discussed. Their properties are also listed. Modified Zagreb indices are introduced and the Zagreb complexity indices reviewed. Their use in QSPR is illustrated by modeling the structure-boiling point relationship Of C(3)-C(8) alkanes using the CROMRsel procedure. The obtained models are in fair agreement with experimental data and are better than many models in the literature. However, in general, the Zagreb indices do not contribute to the best structure-boiling point models of alkanes. Nevertheless, it is interesting to note that the best five-descriptor model that we found in the literature contains the Zagreb M(2) index.

Antimalarial activity of 2,4-diamino-6-quinazoline sulphonamides is modelled using topological as well as quantum chemical descriptors. The results obtained are discussed critically.

The KRAKENX software calculates a large variety of molecular descriptors based on quantum chemistry computations. The program supports over 2000 three-dimensional descriptors that are calculated from the output of different quantum chemistry packages. The current implementation supports semi-empirical MOPAC-based computations and primarily focuses on orientation-independent descriptors that have been discussed in the literature. The descriptor performance has been exemplified using a number of large and diverse datasets and can be seen to produce parsimonious linear models. The software can be run on multiple platforms and is available to academics free of charge.

Life sciences and mathematics are usually considered as quite distant areas of research. But in fact there are close relationships between them, especially in recent years, when computational biology and bioinformatics rapidly evolve. The spectacular developments in the area of biological sciences, particularly those related to sequencing genomes, made evident that an application of formal mathematical and computer science methods is necessary for further discovering the nature of the living world. Among many areas of mathematics being useful in this context, graph theory plays especially important role. It is also worth to remember that, despite the fact that graphs are intensively applied in biology during last three decades, they were used in chemistry (being a basement of molecular biology) more than a century ago. In this chapter a short review of selected applications of labeled graphs in biology and chemistry is given. Some graph theory problems concerning molecules of chemical compounds and DNA sequencing are presented.

The design of materials needed for the storage, delivery, and conversion of (re)useable energy is still hindered by the lack of new, hierarchical molecular screening methodologies that encode information on more than one length scale. Using a molecular network theory as a foundation, we show that in order to describe charge transport in disordered materials the network methodology must be scaled-up. We detail the scale-up through the use of adjacency lists and depth first search algorithms for during operations on the adjacency matrix. We consider two types of electronic acceptors, perylenediimide (PDI) and the fullerene derivative phenyl-C61-butyric acid methyl ester (PCBM), and we demonstrate that the method is scalable to length scales relevant to grain boundary and trap formations. Such boundaries lead to a decrease in the percolation ratio of PDI with system size, while the ratio for PCBM remains constant; further quantifying the stable, diverse transport pathways of PCBM and its success as a charge accepting material.

We study the Randic index for cactus graphs. It is conjectured to be bounded below by radius (for other than an even path), and it is known to obey several bounds based on diameter. We verify the radius bound and strengthen two diameter bounds, both for cacti. Along the way, we produce several other bounds for the Randic index in terms of graph size, order, and valency for several special classes of graphs, including chemical nontrivial cacti.

Valence molecular connectivity indices are based on the concept of valence delta, δ v, that can be derived from general chemical graphs or chemical pseudographs. A general graph or pseudograph has multiple edges and loops and can be used to encode, through the valence delta, chemical entities. Two graph-theoretical concepts derived from chemical pseudographs are the intrinsic (I) and the electrotopological state (E) values, which are the used to define the valence delta of the pseudoconnectivity indices, ψI,S. Complete graphs encode, through a new valence delta, the core electrons of any atoms in a molecule. The connectivity indices, either valence connectivity or pseudoconnectivity, are the starting point to develop the dual connectivity indices. The dual indices show that not only can they assume negative values but also cover a wide range of numerical values. The central parameter of the molecular connectivity theory, the valence delta, defines a completely new set of connectivity indices, which can be distinguished by their configuration and advantageously used to model different properties and activities of compounds.

Let G = (V, E) be an undirected simple graph of order n with m edges without isolated vertices. Further, let d1 ≥ d2 ≥ · · · ≥ dn be vertex degree sequence of G. General Randic index of graph ´ G = (V, E) is defined by Rα = X (i,j)∈E (didj ) α, where α ∈ R − {0}. We consider the case when α = −1 and obtain upper bound for R−1.

The main concern of this article is to present the complete classification of tetracyclic (chemical) graphs and establishing some extremal results with respect to the geometric‐arithmetic index, defined by , where dx denotes the degree of a vertex x in G. In addition, we characterize the extremal graphs of the first and second maximum values of geometric‐arithmetic index of n‐vertex (n ≥ 9) tetracyclic (chemical) graphs. We present the complete classification of tetracyclic (chemical) graphs and establishing the some extremal results with respect to the geometric‐arithmetic index. Also, we characterize the extremal graphs of the first and second maximum values of geometric–arithmetic index of n‐vertex (n ≥ 9) tetracyclic (chemical) graphs.

A large number of medical experiments have confirmed that the features of drugs have a close correlation
with their molecular structure. Drug properties can be obtained by studying the molecular structure
of corresponding drugs. The calculation of the topological index of a drug structure enables
scientists to have a better understanding of the physical chemistry and biological characteristics of
drugs. In this paper, we focus on Hyaluronic Acid-Paclitaxel conjugates which are widely used in the
manufacture of anticancer drugs. Several multiplicative topological indices of this molecular structure
are determined by using of the edge-partition method.

A wide range of new drugs emerge each year with the rapid development of the manufacture of medicines. Once the indicators of these drug molecular structures are calculated in view of defining the topological indices, these can be used to understand their medical properties. In this paper, based on the drug molecular structure analysis, we compute the multiplicative sum connectivity index, the multiplicative Randić index, the multiplicative atomic bond connectivity index, the multiplicative Geometric arithmetic index, the multiplicative Harmonic index and the multiplicative augmented Zagreb index for the molecular structure anticancer drugs SP[n]. Finally, we used Maple 15 to plot surfaces associated to compare our results.

Formic acid (HCOOH) is a suitable hydrogen storage material because of its high gravimetric and volumetric H2 capacities. Although H2 is produced by the thermal decomposition of HCOOH (HCOOH → H2 + CO2, dehydrogenation), the production of water and carbon monoxide (HCOOH → H2O + CO, dehydration) is the major pathway in HCOOH decomposition despite the thermodynamic favorability of the dehydrogenation process over the dehydration process. A large number of experimental and theoretical studies have suggested that both processes are competitive or that the dehydrogenation process has a lower activation energy in HCOOH decomposition. In the present work, we revisit the factors hindering the progress of the dehydrogenation process, using a whole chemical reaction network based on the graph theory. The calculated chemical reaction network shows that the factor controlling the dehydrogenation and dehydration processes is simple and fundamental and can be explained by the oxidation number of carbon and the betweenness centrality. Based on this understanding of the factors hindering the progress of dehydrogenation, the advantage of the dehydration process in HCOOH decomposition is discussed.

The proposed Structural Formula (SF) concept is a version of Graph Theory (GT) with different kinds of vertices and edges. Within SF, any molecule depicted according to the IUPAC rules can be analyzed as it is drawn. The construction of SF requires only a slight modification of the graph definition: a family of sets of vertices and a family of sets of edges are assigned to different kinds of atoms and bonds instead of a single set of vertices and a single set of edges. To easily introduce the physical characteristics of atoms and bonds, we also include a family of weighting functions defined on families of vertices and edges. The characteristics are introduced in analyses of the SF formula through the SF incidence matrix and then, through simple equations, are transferred to other SF matrices, such as Zagreb, Randić and distance and, ultimately, SF topological indices. Finally, we show that the HOMA geometrical aromaticity index can be treated like the SF topological index. © 2019 University of Kragujevac, Faculty of Science. All rights reserved.

Ultimate strength, softening temperature, and water absorption of the polymer composites based on epoxy resin (type ED-20) with unmodified and/or modified by tetraethoxysilane (TEOS) minerals diatomite and andesite are described. Comparison of experimental results obtained for investigated composites shows that the ones containing modified filler have better technical parameters mentioned above than composites with unmodified filler at corresponding loading. It was experimentally shown that the composites containing binary fillers diatomite and andesite at a definite ratio possess optimal characteristics - so called synergistic effect. Experimental results are explained in terms of structural peculiarities of polymer composites. © Aneli J., Mukbaniani O., Markarashvili E., Zaikov G., Klodzinska E., 2013.

Multivariate statistical tools, such as principal-component analysis and multiple-regression analysis, were used for a large number of structural and empirical parameters of some eleven natural amino acids for predicting mutagenicity. An optimized regression model was derived from some selective principal components and solubility data. Plots related to the principal components resulted in the ordination of amino acids as well as the structural and empirical parameters.

A topological index Z is proposed for a connected graph G representing the carbon skeleton of a saturated hydrocarbon. The integer Z is the sum of a set of the numbers p(G,k), which is the number of ways in which such k bonds are so chosen from G that no two of them are connected. For chain molecules Z is closely related to the characteristic polynomial derived from the topological matrix. It is found that Z is correlated well with the topological nature of the carbon skeleton, i.e., the mode of branching and ring closure. Some interesting relations are found, such as a graphical representation of the Fibonacci numbers and a composition principle for counting Z. Correlation of Z with boiling points of saturated hydrocarbons is pointed out.

A procedure is developed for ordering the atoms in a molecule according to the values of their topological characteristics called extended connectivities. The ordering of atoms by means of this procedure in 13 polycyclic hydrocarbons with 2–5 condensed benzenoid rings is compared to the ordering of experimentally determined 1H NMR chemical shifts. Excellent agreement is obtained for about 50 proton chemical shifts in these hydrocarbons, in intramolecular comparisons. Surprisingly, the agreement is better than the results of calculations using the theory of McWeeny.

Four molecular similarity measures have been used to select the nearest neighbor of chemicals in two data sets of 139 hydrocarbons and 15 nitrosamines, respectively. The similarity methods are based on calculated graph invariants which include atom pairs, connectivity indices and information theoretic topological indices. The property of the selected nearest neighbor by each method was taken as the estimate of the property under investigation. The results show that for these data sets, all four methods give reasonable estimates of the properties studied.

One of the fundamental results of Dimensional Analysis is the so‐called Bridgman' s theorem. This theorem states that the only functions that may have dimensional arguments are products of powers of the base quantities of a given system of units. In this work, Bridgman's theorem is discussed and rederived in two different ways, one not involving calculus, and a second one based on a Taylor series expansion analysis.

For a molecular graph G we construct a sequence of graphs L'(G), i = 0, 1, 2,..., such that L°(G) = G, L\G) is the line graph of G, L\G) is the line graph of L\G), etc. Let 6[ be the value of some topological index XI, associated with L'(G). We examine the possibility to model the structural dependency of certain physico-chemical properties of alkanes (boiling point, molar volume, molar refraction, heat of vaporization, critical temperature, critical pressure and surface tension) by means of linear functions of the parameters 6>-, i = 0, 1,..., k. It is shown that the use of line graphs and, in some cases, of the second line graphs, results in a significant improvement in the predictive power of topological indices. The line graphs with f>3 seem to play a negligible role. In the present work the choices for XI with 0-, = number of vertices of Li(G) and θj = Wiener index of Li(G) are pursued.

We initiate here a systematic analysis of molecular descriptors, the quantities used to represent molecules in structure-property and structure-biological activity studies. We illustrate the approach with Hosoya's Z index, which is based on the enumeration of disjoint edges in molecular graphs. The purpose of this analysis is to discern similarities and differences among the molecular descriptors used in regression analysis. In this way, we hope to facilitate the rational use and perhaps the rational design of such descriptors. The main tool in this analysis is the recently described methodology of construction of orthogonalized molecular descriptors in multivariate regression analysis. The analysis points to critical parts of descriptors which are responsible for their performance in a regression.

The descriptive power and utility of molecular connectivity terms derived by a trial-and-error procedure from a medium-sized set of 8 molecular connectivity indices has been tested on different properties of biochemical compounds: amino acids, purines and pyrimidines. The trial-and-error procedure is sometimes based on the entire set of molecular connectivity indices, and sometimes on a subset composed solely of the best molecular connectivity indices. The present paper reports a modeling which encompasses the melting temperatures of natural amino acids, the water solubility of a heterogeneous class of amino acids plus purines and pyrimidines, and three different properties of the DNA-RNA bases (U, T, A, G and C): singlet excitation energies ΔE1 and ΔE2, and molar absorption coefficient ε260. The modeling is in every case very satisfactory, both from the point of view of predictive power and utility of the equations.

The method of linear combinations of connectivity indices has been applied to fit the following physicochemical properties of inorganic molecules: the lattice enthalpies of 32 inorganic compounds, the limiting enthalpies of solution of 12 metal halides, the unfrozen water content, the relative humidity and eutectic temperature of metal chlorides and the unfrozen water content of a mixed set of amino acids and metal chlorides. The achieved fit is very satisfactory and it can be normally attained with the help of a single-index linear combination. The obtained fit draws also a validity test for the delta δv valence number of the inorganic compounds and for a newly defined DZ index based on the number of valence electrons and on the principal quantum number. The fit of the unfrozen water content of the mixed set of amino acids and metal chlorides, achieved by means of linear combinations of reciprocal connectivity indices can be seen as a first hint for the construction of a connectivity model encompassing both organic and inorganic compounds.

A strategy to employ a linear combination of the connectivity indexes (LCCI) method to model a physicochemical property of a series of molecules is outlined throughout. The chosen physicochemical property is the solubility of 19 natural amino acids. This property is modeled with the aid of normal LCCI and of linear combinations of special constructions of connectivty indexes relating to different numbers of amino acids, including and excluding extreme outliers. The employed indexes are analyzed following their descriptive power of solubility of natural amino acids. A linear combination of reciprocal connectivity indexes (LCRCI) showed the best mapping of the water solubility of 16 amino acids while a model based on LCRCI, along with the use of supraConnectivity indexes for the amino acids proline, serine and arginine, achieved very good modeling of the water solubility of the entire set of n = 19 natural amino acids. Linear combinations of fragment connectivty indexes were also analyzed while linear combinations of orthogonal connectivty indexes (LCOCI) were used to improve the modeling and to detect dominant descriptors for this physicochemical property. Non connectivity 'ad hoc' indexes, already used in a previous study with good success, did not achieve the same good quality as the linear combination of reciprocal connectivity indexes.

Computer-assisted methods are applied to the study of the relationship between molecular structure and observed surface tension of small organic alkanes, alkyl esters, and alkyl alcohols. Features of these molecules are encoded using a wide variety of topologic, geometric, and electronic descriptors. The simple correlation between these descriptors and observed surface tension values is examined to gain insight as to which molecular features most influence the observed surface tension. Multivariate linear regression models for each functional group class are also examined. The results of the examination of both the simple correlations and the regression models suggest that molecular surface area is an important feature. The results also show that many descriptors provide surface area information which is specific to particular portions of the molecule, and that this information provides better results in modeling surface tension than the van der Waals or solvent-accessible surface area. Finally, a multiple linear regression model is developed for a combined set of alkanes, alkyl esters, and alkyl alcohols which yields good results for predicting the surface tension of similar compounds.

The simple molecular connectivity indices are analysed for the information in the bond terms. It is found that these terms reflect the relative accessibility of each bond to encounter other bonds of the same molecule in a milieu. The total possibility of each molecule to encounter another molecule in a bimolecular interaction is found to be the molecular connectivity index for that molecule. The molecular connectivity indices are interpreted to be the bimolecular interaction possibilities of a molecule in a milieu.

How molecular structure can be represented mathematically and how this can lead to a better understanding of the connection between molecular structures and properties. Keywords (Audience): Second-Year UndergraduateKeywords (Domain): Organic ChemistryKeywords (Subject): Molecular Properties / Structure

The linear combinations of connectivity indices method (LCCI) is here employed to model the water solubility and activity of 19 natural amino acids. Starting with the molecular connectivity indices, reciprocal and supra molecular connectivity indices are designed to model the solubility and activity spaces of the natural amino acids. The reciprocal and supra molecular reciprocal connectivity indices have been obtained following the variability of the connectivity indices along solubility space of the natural amino acids. A linear combination of the reciprocals of the connectivity indices (LCRCI) showed a satisfactory modelling of the solubility and activity space while a model based on the LCRCI together with the introduction of supra reciprocal molecular connectivity indices for Pro, Ser and Arg achieved an optimal modelling of the solubility and activity space of the natural amino acids.

A novel approach to characterization of heteroatoms in graph theoretical approaches to quantitative structure—activity relationships (QSAR) is outlined. The basis of the approach is the use of diagonal entries of the adjacency matrix as variable parameter, in full analogy to the well known generalization of the Hückel Molecular Orbitals (HMO) method when extended to heteroconjugated systems. The approach is illustrated on clonidine-like compounds where carbon and chlorine atoms are discriminated by using x = −0.20 as the diagonal entry for chlorine atoms. Derived weighted path numbers are used as descriptors and a multiple regression based on three descriptors resulted in the correlation coefficient R = 0.977 and the standard error S = 0.233. This represents a substantial improvement over the best traditional QSAR analysis which involves five descriptors (in a nonlinear correlation equation with R = 0.964 and S = 0.301). A detailed comparison is made with available QSAR results, and the advantages (as well as limitations) of graph theoretical descriptors are discussed.

Molecular connectivity indices have been developed which characterize contributions of neighboring atoms to the CNDO/2 calculated electronic charge of a carbon atom. An analysis of these indices reveals their ability to predict this charge to 0.001 electron.

A molecular connectivity model for the calculation of the relaxation times of amino acids and cyclic dipeptides is presented. A power type regression shows the best fit between experimental and calculated data. The simple molecular connectivity index and the simple sum-delta index are the most appropriate indexes in describing the relaxation times of the investigated compounds. The given computational method shows that it is possible to model tumbling rates of solute molecules in a relatively easy and direct way.

A new topological indexJ (based on distance sumssi as graph invariants) is proposed. For unsaturated or aromatic compounds, fractional bond orders are used in calculatingsi. The degeneracy ofJ is lowest among all single topological indices described so far. The asymptotic behaviour ofJ is discussed, e.g. whenn → ∞ in CnH2n = 2,Jar π for linear alkanes, andJ → ∞ for highly branched ones.

The molecular connectivity functions for Van der Waals constants a and b, Flory's characteristic pressure p∗ and reduced volume Ṽ and the ratio are obtained through regression analysis of data on n-alkanes and isomeric alkanes. Such functions are very useful in predicting the desired property of a compound for which experimental data are lacking. Such connectivity functions, however, fail to describe the properties of mixtures or thermodynamic functions of mixing.Thus, the enthalpies of mixing ΔHM calculated through the use of these connectivity functions, differ largely in sign and/or magnitude from the observed data. Reasons for the discrepancies are discussed.

An efficient algorithm for deriving QSPR/QSAR models with nonorthogonal and ordered orthogonal descriptors, based on orthogonalization of topological indices, is presented. It is applied to structure-boiling point modeling of nonanes as the test case. The selection of the best descriptors from multivariate linear regression modeling is carried out using descriptors which are first orthogonalized. It is shown that such an algorithm is applicable for the selection of the best descriptors in a multivariate linear regression model even to very large sets of descriptors. A computationally-effective method for the (ordered) orthogonalization of topological indices is also introduced. By the use of an ordered orthogonalization procedure it is possible to select the best order of descriptors for orthogonalization. The orthogonalization in the selected order of descriptors produces models with a smaller number of significant descriptors. The comparison between QSPR/QSAR models with nonorthogonal and ordered orthogonal topological indices favors the latter models. It is also shown that the enlarging of the basis set results in better QSPR/QSAR models, but it is not possible to know in advance whether it is better to use a nonlinear transformation of the smaller basis or to compute additional descriptors.

The properties of different connectivity terms derived by a trial-and-error procedure from a medium-sized set of eight connectivity indices have been analysed in relation to their descriptive power and utility. The best connectivity terms normally show dominant features, which allow a reduction in the combinatorial space to be searched, while their δ dependence as well as their particular structure unveil interesting features which can help an understanding of their form. Some of them are based on a regular common pattern, as, e.g. on ratios of parent indices or exponential terms, and sometimes similar terms can show different modeling features. Furthermore, their δ dependence can cover a wide range of possibilities, and unmasks an interesting electronic interpretation of the analysed properties. The many different physicochemical properties modeled by these terms, and in particular by two different families of terms, cover a wide range of classes of compounds: the specific rotations of 64 d- and l-amino acids in three different media, the solubilities of 43 amino acids and purines and pyrimidines, the unfrozen water content of 13 amino acids and inorganic salts, five different properties of the five RNA–DNA bases (U, T, A, G, C), that is, the singlet excitation energies ΔE1 and ΔE2, the oscillation f1 and f2 strengths and the molar absorption coefficient ε260, and the motor octane numbers of 30 alkanes. These modelings with connectivity terms show interesting properties, like, e.g., ΔE1 and ΔE2, modeled by the same term, and further f1, and ε260 (and in a minor way f2), whose modelings are achieved by a term used to model the unfrozen water content, while the modeling of the motor octane numbers uncovers a wide choice of optimal but similar terms. The trial-and-error procedure is sometimes based on the entire set of connectivity indices, and sometimes on a subset composed by the connectivity indices of the best linear combination. The modeling of the specific rotation of amino acids in different media introduces an expansion test, which consists in using the best descriptive terms for specific rotations of amino acids in aqueous and `expand' their use to model the same property for d- and l-amino acids in acidic and basic solutions.

In recent years topological indices have come prominently to the fore on several different fronts. The prime reason for the wide publicity accorded these indices has been the remarkable ability they possess to correlate and predict the properties of a vast range of molecular species. However, although the indices are now seemingly ubiquitous in the chemical context, they are subject to a number of constraints on their realm of applicability. Here we explore these constraints and attempt to draw realistic boundaries within which the indices may be legitimately used. It is noted in passing that current uses of the indices far from exhaust their manifold future possibilities. Mention is made of some of the numerous promising areas of application still awaiting development.

The beginning student of organic chemistry is often bewildered by what appears to be an enormous maze of random structural variations and reactions that can be mastered only by tedious memorization. To the organic chemist, however, the same subject is often a beautifully ordered discipline of elegant simplicity. An important value of learning organic chemistry is the mastering of “organic thinking,” an approach to intellectual processing whereby the “sameness” of many families of structures and reactions is revealed. This article offers the author's personal views of organic thinking and explores the intellectual and scientific foundations of organic chemistry and of the powerful methods that provide the field with a platform for making rapid conceptual and experimental advances. It is proposed that these methods involve a geometric and topological approach to scientific reasoning within the framework of scientific paradigms that guide experimental design and execution. The basis of this approach is considered in relation to day-to-day thinking, problem solving, and the psychological drive for intellectual closure. The power of the approach is illustrated by the analysis of several photochemical and chemiluminescent reactions.

Molecular connectivity has been used to find new β-blocker drugs using linear discriminant analysis and connectivity functions with different topological descriptors. Among the selected compounds stands out the probucol and the β-carotene. Both of them interact with β adrenoceptors.

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A quest for an optimal missing descriptor is started here, a descriptor that can include information on van der Waals and hydrogen bond interactions. This quest is centered on the construction of special molecular connectivity terms that make use of an empirical parameter, the dielectric constant, and three different “ad hoc” parameters indirectly related to the dielectric constant that should render molecular connectivity terms able to describe also noncovalent interactions. The molar mass M was also used, as an empirical parameter, to derive molecular connectivity terms strongly related to the mass of the compounds. Derived semiempirical molecular connectivity terms were used to model 11 different properties of a wide heterogeneous range of solvents, for which χ indices alone and their linear combinations are insufficient descriptors: boiling points, melting points, refractive indices, density, viscosities, flashpoint values, elutropic values, UV cutoff values, dipolar moments, magnetic susceptibility, and the same dielectric constant. Achieved modelings are quite fair, and with the exclusion of the refractive indices all of them needed semiempirical terms, mainly based on supramolecular connectivity indices of different forms. Further, a common term, within the “double sieve” approximation for the melting points, has been detected for the boiling and melting temperatures. Two types of valence molecular connectivity indices have been used all along the modeling: χv indices based on δv valence and χz indices based on δz valence numbers. This last kind of χz index plays an important role in X terms for the flashpoints and melting points only.

The descriptive and utility power of linear combinations of special construction of connectivity indices (LCXCI) derived by a trial-and-error procedure from a medium-sized set of eight connectivity indices or from a subset of it has been tested on several properties of different classes of bioorganic and inorganic compounds. Two techniques have been tested to choose the appropriate combination of indices: the forward selection and the complete combinatorial technique. While the latter searches the entire combinatorial space and the first searches only a subspace of it, this last, nevertheless, has many advantages among which to be a good tool for an elementary and direct test for newly defined indices. The modeling of the side-chain volume V of 18 amino acids (AA) is perfectly achieved by a composite index together with a connectivity index, and, while the modeling of pI of 21 amino acids is satisfactorily accomplished by special 0χv-fractional indices that are also rather good descriptors of the melting Tm points of 20 amino acids, the solubility S of 16 and 20 amino acids is nicely described by reciprocal and suprareciprocal connectivity indices, respectively, a description that seems to have nothing in common with the modeling of the same property for 23 purines and pyrimidines (PP) achieved by squared supraconnectivity indices. Nevertheless, the modeling of the solubility of the entire heterogeneous class of n = 43 amino acids, purines, and pyrimidines could be satisfactorily achieved with a set of supracomposite indices based on the χtv index mainly. The modeling of the motor octane MON number of 30 alkanes and of the melting points of 17 and 14 alkanes shows how far a minimum set of four connectivity indices can positively replace a larger set of 17 indices, while the modeling of the lattice ΔHLφ enthalpies of 20 metal halides by a mixed set of normal and composite indices introduces and stimulates the problem of the definition of a connectivity model for inorganic compounds. The utility of the given LCXCI is generally rather high as many properties can be satisfactorily modeled by one or just two indices (V, pI, S(AA), S(PP), and ΔHLφ) and it can be enhanced, especially when the modeling requires more than three or four indices, with the introduction of the corresponding orthogonal indices.

Eight physical properties (boiling points, molar volumes, molar refractions, heats of vaporization, surface tensions, melting points, critical temperatures, and critical pressures) of 74 normal and branched alkanes were examined by molecular modeling techniques. Structural parameters employed include Wiener indices, connectivity indices, ad hoc descriptors, information indices, and molecular volumes and surface areas. Most of the properties were well modeled (rÂ² > 0.97) by the Wiener indices, connectivity indices, and ad hoc descriptors. An exception was the melting points, which were not well modeled by any of the available indices. Factor analysis (principal component analysis) was used to examine the intrinsic dimensionalities of the data and parameter sets. A single factor accounts for about 82% of the variance in the eight physical properties, two factors account for 94%, and three factors account for 99%. The melting points load strongly on a factor independent of the other properties. Of the examined parameter sets, the connectivity indices exhibited the highest dimensionality.

The London equation for dispersion energy between nonpolar molecules is used to calculate boiling point as a function of polarizability, ionization energy, and size, provided the shape of the molecule is taken into account. For spherical molecules, density is proportional to molecular weight M divided by the cube of the radius. For the zero-point energy hv0, the ionization potential I is used. For polarizability, the molar refraction RM is used and the boiling point Tb, is assumed proportional to the energy of vaporization (Trouton's rule). The result is that Tb1/2 is proportional to RMI1/2/ Vb, where Vb is the molar volume at Tb. The data for rare gases and for group 4A tetrahalides give good straight lines. The method is then extended to cylindrical and flat molecules, in which case Tb1/2 is proportional to RMI1/2L3/2/Vb3/2, where L is the length of the molecule, and RMI1/2A3/Vb3, where A is the area of the molecule. Good straight lines are obtained for normal hydrocarbons (cylindrical) and aromatic compounds (flat).

A molecular connectivity model for the calculation of some physicochemical properties of α-amino acids such as molecular weights, molecular volumes, isoelectric points, relaxation times, crystal densities, solubilities, specific rotations, and atom charges is presented. The given model, based on a set of six molecular connectivity indexes, MCI = {D, Dv, 0X, 0Xv, 1X, 1Xv} gives a good description of all these physicochemical properties, except atom charges. While calculation of isoelectric points and of solubilities requires a connectivity model centered on the connectivity indexes of the functional fragments of the α-amino acids, other properties, with the exception of the atomic charges, are centered on the molecular connectivity indexes of the whole amino acid. A subset composed of three indexes from this set gives also a rather good description of these properties. The molecular connectivity model of the atom charges shows a simple linear correlation between atomic δv (and δ for sp3 carbons) index and atom charges, broadening thus the meaning of delta and valence delta numbers used to describe the given set of physicochemical properties in α-amino acids.

The molecular connectivity index combinations used to describe the physicochemical properties of the alpha-amino acids were specifically analyzed in terms of their statistical meaning and, especially, of their Q (quality) value. Combinations of ordinary connectivity indexes provide a good description of nearly every physicochemical property examined but especially of the isoelectric point, side-chain volume, and crystal density of the alpha-amino acids. Some features of these combinations of connectivity indices are as follows: (i) the combination with the maximum number of molecular connectivity indexes does not always show the best quality; (ii) successive inclusion of the next good index to the best combination of indexes does not always produce the next best combination of indexes, and (iii) even highly intercorrelated indexes can contribute to the best combination of indexes. It was also detected that valence molecular connectivity indexes by themselves provide a rather good description of nearly every property with the exception of specific rotation of amino acids, which is better described by a set of two simple connectivity indexes. Orthogonal indexes, obtained from the ordinary molecular connectivity indexes, show in some cases the best single-index regressions and the best multi-index description of the molecular weights and relaxation rates. The best description of the specific rotation and solubility of the alpha-amino acids is, instead, achieved by the aid of the nonconnectivity Kidera, Konishi, Oka, Ooi; and Scheraga indexes.

The modeling power of the method of linear combinations of connectivity indexes (LCCI), based on a minimal and on an expanded set of connectivity indexes, has been tested on several properties of different classes of organic compounds: the melting points and motor octane numbers of alkanes, the melting points and solubilities of caffein homologues, and four different physicochemical properties of organophosphorus compounds. The modeling of the first property, a classical shape-dependent property and up to date a challenging problem of molecular modeling, was resolved by partitioning the entire set of alkanes into congruent subsets. A minimal set of normal and valence connectivity indexes was able to model the melting points of caffein homologues that have quite similar molecular shapes and sizes, while the modeling of the solubilities of these homologues was unravelled by taking into consideration their association in solution and by employing linear combinations of squared connectivity indexes. The very effective modeling of the two different types (shape- and size-dependent) of properties of the organophosphorus compounds, with a minimal set of connectivity indexes, delineates also a test for the proposed valence delta(v) value of phosphorus in organophosphorus derivatives. Linear LCOCI combinations of orthogonal connectivity indexes were also tested to improve, if possible, the modeling of the properties of the given classes of compounds. Modeled properties show that the connectivity indexes can be highly dependent on the detailed knowledge of the physicochemical state of the investigated system and that, usually, LCCIs with a minimal basis set yield quite adequate modeling.

A discussion of Franck-Condon factors and the associated Condon parabola suitable for inclusion in a junior- or senior-level physical chemistry or quantum physics course is presented. The Condon parabola is a convenient means of displaying and visualizing the values of Franck-Condon factors for transitions between two electronic states of a diatomic molecule. In the Condon parabola, Franck-Condon factors are displayed using a contour format to better bring out their parabola-like distribution and the associated interference pattern corresponding to transition intensities. A description of the Condon parabola and Franck-Condon factors for a model in which harmonic oscillators are used to represent the electronic states of a diatomic molecule is given. The effects of varying the equilibrium internuclear separations of the states and the state force constants are illustrated. A Condon parabola for a real diatomic molecule, diatomic potassium, is also shown, and similarities to the harmonic oscillator model are discussed. Keywords (Audience): Upper-Division Undergraduate

This paper is a rebuttal of the suggestion that the term "metallic bond" be stricken from the chemist's lexicon. The argument of the critics of current usage is that the metallic bond is simply a special case of the covalent bond and that it can be discussed adequately in terms of molecular orbital theory. They define a "metal" as a substance with high electronic conductivity. The rebuttal points out that the historical definition of a metal has (for millennia) been in terms of the ductility of a substance - a property not possessed by "organic" and "ceramic" electronic conductors. This complex and unique property is supported by the characteristics of the metallic bond. In particular, it requires a sufficiently high concentration of nearly free electrons to form a Fermi liquid because ductility requires the bonding to be as nondirectional as possible. This is the antithesis of covalent bonding, as in diamond, for example. Keywords (Audience): General Public

This paper attempts to introduce the reader to the topographical index. Keywords (Audience): Upper-Division Undergraduate

New topological descriptors, namely ''charge indexes'', are presented in this paper. Their ability for the description of the molecular charge distribution is established by correlating them with the dipole moment of a heterogeneous set of hydrocarbons, as well as with the boiling temperature of alkanes and alcohols and the vaporization enthalpy of alkanes. Moreover, it is clearly demonstrated that this ability is higher than that shown by the chi connectivity and Wiener indexes.

The normal boiling points for olefins are predicted by use of exclusively topological descriptors derived from molecular structure. Predictive equations having from one to eight independent variables were obtained by applying multiple linear regression analysis to a set of topological descriptors (independent variables) and the observed boiling points of 123 Câ-Cââ olefins (dependent variable). The best model found, which included eight descriptors, yielded a correlation coefficient of 0.999 and an estimated standard deviation of 1.78 Â°C.

We report on optimal molecular connectivity descriptors for nitrogen atoms in amines for use in structure–property correlations. The descriptors represent generalized molecular connectivity indices with adjusted diagonal entries in the adjacency matrices of the corresponding molecular graphs, such that the standard error in a regression for boiling points in a set of amines is minimized. Advantages of the so-optimized descriptors for multivariate regression analysis in structure–property–activity studies are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 1209–1215, 1998

The physical basis for valence molecular connectivity was studied. The δv and δ values are cardinal numbers describing the electronic structure of atoms in their valence states. The value δv + δ describes the volume of a bonding atom while the value δv − δ describes the electronegativity. By using the principle of electronegativity equalization, bond electronegativity is defined as (δviδvj)−1/2, and the valence molecular connectivity index (1χv) is derived as a sum of these bond descriptions. The valence chi index is interpreted in terms of the information encoded, describing both the volume and electronic characteristics of bonds in molecules. Examples of close relationships with molecular volume and electronic properties are shown. A new way of estimating valence state electronegativity is proposed from a count of exterjacent electrons divided by the quantum number squared for at least the first three quantum levels.

A numerical index is proposed that ranks solvents according to their polarity. It is based entirely on structure, encoding the relative content of exterjacent electrons in the molecule. The index is the first-order valence molecular connectivity index, 1Xv The index is modified for the number of isolated functional groups in the molecule. A comparison with solvent polarity indexes based on several experimental methods reveals a good relationship. The polarity index proposed can be quickly calculated, it does not depend on the availability of the actual molecule, and it permits prediction of solvent polarity or the polarity of mixtures.

The connectivity index, easily computed by arithmetic and based upon the degree of connectedness at each vertex in the molecular skeleton, is shown to give highly significant correlations with water solubility of branched, cyclic, and straight-chain alcohols and hydrocarbons as well as with boiling points of alcohols. These correlations are superior to those based on well-founded theory relating to solvent cavity surface area. An empirical modification to the connectivity index gave an improved correlation for both solubilities and boiling points.

A semiempirical SCF LCAO MO-CI calculation has been performed for the nucleotide bases. According to the results we obtain a better agreement with the experimental singlet excitation energies, if we use the SCF eigenvectors of the Pople-type matrix in the CI calculation instead of the eigenvectors obtained after the first iteration step. On the basis of some parameter variation a set of integrals has been determined which yields as largest deviation between the experimental and theoretical excitation energies for the first two intensive excitations of the five bases (A, T, G, C and U) 1 eV. The possibilities of further improvements are discussed.
In the second part of the calculations the oscillator strength values (f) of the G-C base pair and of the GpG, GpC, CpG and CpC dinucleotides have been determined using the first 16 singlet excited configurations of these composite systems for the CI calculation. The comparison of the results obtained with the appropriatef values of the constituent single bases shows a hypochromicity of the first absorption band system in the case of all the four dinucleotides and a slight hyperchromicity in the case of the G-C base pair.

Topological indices (TIs) have been used to study structure-activity relationships (SAR) with respect to the physical, chemical, and biological properties of congeneric sets of molecules. Since there are many TIs and many are correlated, it is important that we identify redundancies and extract useful information from TIs into a smaller number of parameters. Moreover, it is important to determine if TIs, or parameters derived from TIs, can be used for global SAR models of diverse sets of chemicals. We calculated seventy-one TIs for three groups of molecules of increasing complexity and diversity: (a) 74 alkanes, (b) 29 alkylbenzenes, and (c) 37 polycyclic aromatic hydrocarbons (PAHs). Principal components analysis (PCA) revealed that a few principal components (PCs) could extract most of the information encoded by the seventy-one TIs. The structural basis of the first few PCs could be derived from their pattern of correlation with individual TIs. For the three sets of molecules, viz. alkanes, alkylbenzenes and PAHs, PCs were able to predict the boiling points reasonably well. Also, for the combined set of 140 chemicals consisting of the alkanes, alkylbenzenes and PAHs, the derived PCs were not as effective in predicting properties as in the case of individual classes of compounds.

The descriptive and utility power of linear combinations of connectivity terms (LCCT) derived by a trial-and-error procedure from a medium-sized set of 8 connectivity indices: {χ} = {D, Dv,0χ,0χ
v,1χ,1χ
v,χ
t,χ
tv} or from a subset of it has been tested on properties of heterogeneous classes of biochemical compounds centered on the homogeneous class of natural L-amino acids. To choose the appropriate combination of indices the forward selection and the complete combinatorial technique have been used, whenever more than a single term was necessary for the description. The forward selection technique searches only a subspace of the complete combinatorial space, but nevertheless has many advantages among which to be a good tool for an elementary and direct test for newly defined indices. The modeling has been followed centering the attention not only on the predictive power of the proposed linear equations but also on their utility. The modeling of the solubility of the entire heterogeneous class of n = 43 amino acids, purities and pyrimidines could satisfactorily be achieved with a set of supraconnectivity terms based on theχ
tv index mainly. The unfrozen water content of a mixed class of inorganic salts and natural amino acids has satisfactorily been modeled with two connectivity terms and the modeling shows a remarkable utility. The utility of the given LCCT can nevertheless be enhanced, especially when the modeling requires 2 or more terms, with the introduction of the corresponding orthogonal indices, as can be seen for S(AA + PP) and UWC.
Further, theδ cardinal number is used as starting point for the definition of a supravalence index Δ to be used for a topological codification of the genetic code and the amino acids in proteins. In fact, the notion of supravalence can be extended to the triplet code words to generate the different families and subfamilies of the genetic code and to visualize the connections of amino acids in proteins. Three properties of the DNA-RNA bases (U, T, A, G and C), the singlet excitation energies ΔE1 and δE2, and the molar absorption coefficient ε260 have been simulated with a single connectivity term chosen from the same medium-sized set of 8 molecular connectivity indices.

Structural invariants used currently as molecular descriptors in structure-property (SPR) and structure-activity relationship (SAR) studies are based on molecular graphs rather than on a structure as a 3-dimensional (3-D) object. Here we outline an approach which extends graph-theoretical methodology to structures embedded in 3-D space. Derived descriptors are sensitive to the 3-D characteristics of a molecular structure and change with variations in molecular conformations. We illustrate the approach by examining conformations of smaller alkanes, including selected conformations of cyclohexane, methyl-cyclohexanes and cyclooctane. In particular, we report the atomic path numbers and atomic ID numbers (i.e. local descriptors), and molecular path numbers and molecular ID numbers (i.e. global descriptors). The path numbers were weighted so that a path of length 1 corresponds to the molecular connectivity index χ.This paper should correct the widely held misconception that graph-theoretical analysis is not applicable to 3-D molecular structures, and at the same time should alert those interested in SPR and SAR to the availability of 3-D structural invariants of potential use in QSPR and QSAR models. The examples given and the computational background suffice to inform a potential user to write her/his own software, or to modify the existing ALL PATH program in order to obtain the novel 3-D molecular invariants. The computations were performed on an Apple IIe personal computer (with a program written in BASIC).